d-Disjunct matrices: bounds and Lovász Local Lemma Original Research Article

A binary matrix is said to be d-disjunct if the union (or Boolean sum) of any d columns does not contain any other column. Such matrices constitute a basis for nonadaptive group testing algorithms and binary d-superimposed codes. Let t(d,n) denote the minimum number of rows for a d-disjunct matrix with n columns. In this note we study the bounds of t(d,n) and its variations. Lovász Local Lemma (Colloq. Math. Soc. Jãnos Bolyai 10 (1974) 609–627; The Probabilistic Method, Wiley, New York, 1992 (2nd Edition, 2000)) and other probabilistic methods are used to extract better bounds. For a given random t×n binary matrix, the Stein–Chen method is used to measure how ‘bad’ it is from a d-disjunct matrix.